Download Economics of Space Exploration Bill Gibson UVM Fall 2010

Can Humans Fly in Space
Costs
Economics of Space Exploration
Bill Gibson
UVM Fall 2010
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Flight in Space would not be economically possible without
orbits
Without orbits, fuel needed for sustained flight is impossibly
heavy.
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Flight in Space would not be economically possible without
orbits
Without orbits, fuel needed for sustained flight is impossibly
heavy.
Spacecraft cannot operate like airplanes
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Flight in Space would not be economically possible without
orbits
Without orbits, fuel needed for sustained flight is impossibly
heavy.
Spacecraft cannot operate like airplanes
Flights would have to be very short.
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Flight in Space would not be economically possible without
orbits
Without orbits, fuel needed for sustained flight is impossibly
heavy.
Spacecraft cannot operate like airplanes
Flights would have to be very short.
Newton’s first law required;
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Flight in Space would not be economically possible without
orbits
Without orbits, fuel needed for sustained flight is impossibly
heavy.
Spacecraft cannot operate like airplanes
Flights would have to be very short.
Newton’s first law required;
Body in motion continues in motion except when acted upon
by an external force.
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Flight in Space would not be economically possible without
orbits
Without orbits, fuel needed for sustained flight is impossibly
heavy.
Spacecraft cannot operate like airplanes
Flights would have to be very short.
Newton’s first law required;
Body in motion continues in motion except when acted upon
by an external force.
Example
Can we use gravity as a way of getting from here to there in space?
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Flight in Space would not be economically possible without
orbits
Without orbits, fuel needed for sustained flight is impossibly
heavy.
Spacecraft cannot operate like airplanes
Flights would have to be very short.
Newton’s first law required;
Body in motion continues in motion except when acted upon
by an external force.
Example
Can we use gravity as a way of getting from here to there in space?
Answer: Yes!
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Orbits
After Sputnik 1 4 October 1957
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Orbits
After Sputnik 1 4 October 1957
First U.S. 31 January 1958 Explorer I
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Orbits
After Sputnik 1 4 October 1957
First U.S. 31 January 1958 Explorer I
USA: TIROS 1 weather sat. 1 April 60
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Orbits
After Sputnik 1 4 October 1957
First U.S. 31 January 1958 Explorer I
USA: TIROS 1 weather sat. 1 April 60
Human orbit: Yuri Gagarin 12 April 61
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Orbits
After Sputnik 1 4 October 1957
First U.S. 31 January 1958 Explorer I
USA: TIROS 1 weather sat. 1 April 60
Human orbit: Yuri Gagarin 12 April 61
First US human orbit: John Glenn 20 Feb 62
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Orbits
After Sputnik 1 4 October 1957
First U.S. 31 January 1958 Explorer I
USA: TIROS 1 weather sat. 1 April 60
Human orbit: Yuri Gagarin 12 April 61
First US human orbit: John Glenn 20 Feb 62
TV Sat. Telstar I 10 July 62
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Orbits
After Sputnik 1 4 October 1957
First U.S. 31 January 1958 Explorer I
USA: TIROS 1 weather sat. 1 April 60
Human orbit: Yuri Gagarin 12 April 61
First US human orbit: John Glenn 20 Feb 62
TV Sat. Telstar I 10 July 62
Humans in lunar orbit 24 Dec 68, Anders, Lovell and Borman
(Apollo 8)
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Orbits
After Sputnik 1 4 October 1957
First U.S. 31 January 1958 Explorer I
USA: TIROS 1 weather sat. 1 April 60
Human orbit: Yuri Gagarin 12 April 61
First US human orbit: John Glenn 20 Feb 62
TV Sat. Telstar I 10 July 62
Humans in lunar orbit 24 Dec 68, Anders, Lovell and Borman
(Apollo 8)
Example
What is the most recent orbital flight of great interest?
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Orbits
After Sputnik 1 4 October 1957
First U.S. 31 January 1958 Explorer I
USA: TIROS 1 weather sat. 1 April 60
Human orbit: Yuri Gagarin 12 April 61
First US human orbit: John Glenn 20 Feb 62
TV Sat. Telstar I 10 July 62
Humans in lunar orbit 24 Dec 68, Anders, Lovell and Borman
(Apollo 8)
Example
What is the most recent orbital flight of great interest?
Answer: SpaceX...may fundamentally change US space policy
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Basics
meter 39 inches meter per second 2.236 mph; km/sec 2,236
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Basics
meter 39 inches meter per second 2.236 mph; km/sec 2,236
Mach: speed of sound; depends on altitude and pressure.
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Basics
meter 39 inches meter per second 2.236 mph; km/sec 2,236
Mach: speed of sound; depends on altitude and pressure.
Throw ball from tall building: ball lands on ground in time t.
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Basics
meter 39 inches meter per second 2.236 mph; km/sec 2,236
Mach: speed of sound; depends on altitude and pressure.
Throw ball from tall building: ball lands on ground in time t.
Throw ball harder and lands on ground farther but in same
time t.
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Basics
meter 39 inches meter per second 2.236 mph; km/sec 2,236
Mach: speed of sound; depends on altitude and pressure.
Throw ball from tall building: ball lands on ground in time t.
Throw ball harder and lands on ground farther but in same
time t.
Acceleration due to gravity 10m/sec2 constant
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Basics
meter 39 inches meter per second 2.236 mph; km/sec 2,236
Mach: speed of sound; depends on altitude and pressure.
Throw ball from tall building: ball lands on ground in time t.
Throw ball harder and lands on ground farther but in same
time t.
Acceleration due to gravity 10m/sec2 constant
Example
Before Newton, the bible said there were angels behind heavily
bodies pushing them along their orbits. What could Newton
contribute to this explanation?
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Basics
meter 39 inches meter per second 2.236 mph; km/sec 2,236
Mach: speed of sound; depends on altitude and pressure.
Throw ball from tall building: ball lands on ground in time t.
Throw ball harder and lands on ground farther but in same
time t.
Acceleration due to gravity 10m/sec2 constant
Example
Before Newton, the bible said there were angels behind heavily
bodies pushing them along their orbits. What could Newton
contribute to this explanation?
Answer: Newton rotated the angels 90 degrees! (R.
Feynman)
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Orbit
Gravity depends only on altitude; pulls only downward
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Orbit
Gravity depends only on altitude; pulls only downward
Now throw ball hard enough that during the time it takes to
fall one meter, the curvature of the earth drops away by one
meter.
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Orbit
Gravity depends only on altitude; pulls only downward
Now throw ball hard enough that during the time it takes to
fall one meter, the curvature of the earth drops away by one
meter.
Ball will not hit the ground at all
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Orbit
Gravity depends only on altitude; pulls only downward
Now throw ball hard enough that during the time it takes to
fall one meter, the curvature of the earth drops away by one
meter.
Ball will not hit the ground at all
Will follow circular orbit.
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Orbit
Gravity depends only on altitude; pulls only downward
Now throw ball hard enough that during the time it takes to
fall one meter, the curvature of the earth drops away by one
meter.
Ball will not hit the ground at all
Will follow circular orbit.
Object will not gain or loose altitude relative to the earth
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Orbit
Gravity depends only on altitude; pulls only downward
Now throw ball hard enough that during the time it takes to
fall one meter, the curvature of the earth drops away by one
meter.
Ball will not hit the ground at all
Will follow circular orbit.
Object will not gain or loose altitude relative to the earth
Example
With no fuel, what keeps object in orbit?
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Orbit
Gravity depends only on altitude; pulls only downward
Now throw ball hard enough that during the time it takes to
fall one meter, the curvature of the earth drops away by one
meter.
Ball will not hit the ground at all
Will follow circular orbit.
Object will not gain or loose altitude relative to the earth
Example
With no fuel, what keeps object in orbit?
Answer: Gravity and Newton’s First Law: Gravity bends
trajectory of object.
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Orbits
If energy mv 2 /2 were greater than work done by gravity in
bending the trajectory, then orbit is elliptical
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Orbits
If energy mv 2 /2 were greater than work done by gravity in
bending the trajectory, then orbit is elliptical
Gravity catches continues to bend the trajectory however,
slowing the object
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Orbits
If energy mv 2 /2 were greater than work done by gravity in
bending the trajectory, then orbit is elliptical
Gravity catches continues to bend the trajectory however,
slowing the object
Orbit reaches its apogee when the bending force is
perpendicular to the velocity.
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Orbits
If energy mv 2 /2 were greater than work done by gravity in
bending the trajectory, then orbit is elliptical
Gravity catches continues to bend the trajectory however,
slowing the object
Orbit reaches its apogee when the bending force is
perpendicular to the velocity.
This is slowest point in the orbit
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Orbits
If energy mv 2 /2 were greater than work done by gravity in
bending the trajectory, then orbit is elliptical
Gravity catches continues to bend the trajectory however,
slowing the object
Orbit reaches its apogee when the bending force is
perpendicular to the velocity.
This is slowest point in the orbit
One second later the bending forces is accelerating the object
back toward the earth, bending force increases (as object gets
closer)
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Orbits
If energy mv 2 /2 were greater than work done by gravity in
bending the trajectory, then orbit is elliptical
Gravity catches continues to bend the trajectory however,
slowing the object
Orbit reaches its apogee when the bending force is
perpendicular to the velocity.
This is slowest point in the orbit
One second later the bending forces is accelerating the object
back toward the earth, bending force increases (as object gets
closer)
Example
if gravity all of a sudden disappears, what would happen to object:
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Orbits
If energy mv 2 /2 were greater than work done by gravity in
bending the trajectory, then orbit is elliptical
Gravity catches continues to bend the trajectory however,
slowing the object
Orbit reaches its apogee when the bending force is
perpendicular to the velocity.
This is slowest point in the orbit
One second later the bending forces is accelerating the object
back toward the earth, bending force increases (as object gets
closer)
Example
if gravity all of a sudden disappears, what would happen to object:
Answer: It would fly away from the earth on a linear trajectory.
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Escape velocity
Bending force increases but so does energy of the orbital
object.
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Escape velocity
Bending force increases but so does energy of the orbital
object.
If bending force wins, then object follows elliptical path until
it reaches perigee, the point at which the bending forces is
again perpendicular to the velocity.
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Escape velocity
Bending force increases but so does energy of the orbital
object.
If bending force wins, then object follows elliptical path until
it reaches perigee, the point at which the bending forces is
again perpendicular to the velocity.
This is the fastest point in the orbit; but attraction of gravity
is also the strongest so the object remains in orbit.
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Escape velocity
Bending force increases but so does energy of the orbital
object.
If bending force wins, then object follows elliptical path until
it reaches perigee, the point at which the bending forces is
again perpendicular to the velocity.
This is the fastest point in the orbit; but attraction of gravity
is also the strongest so the object remains in orbit.
If the energy of object is greater than the work done by
gravity, the object escapes orbit.
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Escape velocity
Bending force increases but so does energy of the orbital
object.
If bending force wins, then object follows elliptical path until
it reaches perigee, the point at which the bending forces is
again perpendicular to the velocity.
This is the fastest point in the orbit; but attraction of gravity
is also the strongest so the object remains in orbit.
If the energy of object is greater than the work done by
gravity, the object escapes orbit.
7,905 m/s (Earth) 1,680 m/s (Lunar)
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Escape velocity
Bending force increases but so does energy of the orbital
object.
If bending force wins, then object follows elliptical path until
it reaches perigee, the point at which the bending forces is
again perpendicular to the velocity.
This is the fastest point in the orbit; but attraction of gravity
is also the strongest so the object remains in orbit.
If the energy of object is greater than the work done by
gravity, the object escapes orbit.
7,905 m/s (Earth) 1,680 m/s (Lunar)
Example
Do orbital velocities depend on altitude?
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Escape velocity
Bending force increases but so does energy of the orbital
object.
If bending force wins, then object follows elliptical path until
it reaches perigee, the point at which the bending forces is
again perpendicular to the velocity.
This is the fastest point in the orbit; but attraction of gravity
is also the strongest so the object remains in orbit.
If the energy of object is greater than the work done by
gravity, the object escapes orbit.
7,905 m/s (Earth) 1,680 m/s (Lunar)
Example
Do orbital velocities depend on altitude?
Answer: Yes! Gravity is weaker at higher altitudes.
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Strength of Earth’s Gravity–diminishes with square of
altitude
Gravity and Altitude
Sea level
10 km
100 km
1000 km
10,000 km
343,400 km (L1)
100%
99.7%
97
74.7
15.17
0
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Gravity: to derive the 5 meters in one second write
dv
m
= 9.8 2
dt
s
where a = 9.8 sm2
Z
dv
v
v
m
dt
s2
m
= 9.8 2 t
s
m
= 9.8 t
Z s
= 9.8
Z
m
x =
vdt = 9.8
s
m t2
x = 9.8
s 2
m1
= 4.9m
x = 9.8
s 2
Bill Gibson
Z
tdt
University of Vermont
Can Humans Fly in Space
Costs
Orbit
Gravity depends only on altitude; pulls only downward
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Orbit
Gravity depends only on altitude; pulls only downward
Now throw ball hard enough that during the time it takes to
fall one meter, the curvature of the earth drops away by one
meter.
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Orbit
Gravity depends only on altitude; pulls only downward
Now throw ball hard enough that during the time it takes to
fall one meter, the curvature of the earth drops away by one
meter.
Ball will not hit the ground at all
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Orbit
Gravity depends only on altitude; pulls only downward
Now throw ball hard enough that during the time it takes to
fall one meter, the curvature of the earth drops away by one
meter.
Ball will not hit the ground at all
Will follow circular orbit.
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Orbit
Gravity depends only on altitude; pulls only downward
Now throw ball hard enough that during the time it takes to
fall one meter, the curvature of the earth drops away by one
meter.
Ball will not hit the ground at all
Will follow circular orbit.
Object will not gain or loose altitude relative to the earth
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Orbit
Gravity depends only on altitude; pulls only downward
Now throw ball hard enough that during the time it takes to
fall one meter, the curvature of the earth drops away by one
meter.
Ball will not hit the ground at all
Will follow circular orbit.
Object will not gain or loose altitude relative to the earth
Example
With no fuel, what keeps object in orbit?
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Orbit
Gravity depends only on altitude; pulls only downward
Now throw ball hard enough that during the time it takes to
fall one meter, the curvature of the earth drops away by one
meter.
Ball will not hit the ground at all
Will follow circular orbit.
Object will not gain or loose altitude relative to the earth
Example
With no fuel, what keeps object in orbit?
Answer: Gravity and Newton’s First Law: Gravity bends
trajectory of object.
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Orbits
If energy mv 2 /2 were greater than work done by gravity in
bending the trajectory, then orbit is elliptical
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Orbits
If energy mv 2 /2 were greater than work done by gravity in
bending the trajectory, then orbit is elliptical
Gravity catches continues to bend the trajectory however,
slowing the object
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Orbits
If energy mv 2 /2 were greater than work done by gravity in
bending the trajectory, then orbit is elliptical
Gravity catches continues to bend the trajectory however,
slowing the object
Orbit reaches its apogee when the bending force is
perpendicular to the velocity.
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Orbits
If energy mv 2 /2 were greater than work done by gravity in
bending the trajectory, then orbit is elliptical
Gravity catches continues to bend the trajectory however,
slowing the object
Orbit reaches its apogee when the bending force is
perpendicular to the velocity.
This is slowest point in the orbit
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Orbits
If energy mv 2 /2 were greater than work done by gravity in
bending the trajectory, then orbit is elliptical
Gravity catches continues to bend the trajectory however,
slowing the object
Orbit reaches its apogee when the bending force is
perpendicular to the velocity.
This is slowest point in the orbit
One second later the bending forces is accelerating the object
back toward the earth, bending force increases (as object gets
closer)
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Orbits
If energy mv 2 /2 were greater than work done by gravity in
bending the trajectory, then orbit is elliptical
Gravity catches continues to bend the trajectory however,
slowing the object
Orbit reaches its apogee when the bending force is
perpendicular to the velocity.
This is slowest point in the orbit
One second later the bending forces is accelerating the object
back toward the earth, bending force increases (as object gets
closer)
Example
if gravity all of a sudden disappears, what would happen to object:
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Orbits
If energy mv 2 /2 were greater than work done by gravity in
bending the trajectory, then orbit is elliptical
Gravity catches continues to bend the trajectory however,
slowing the object
Orbit reaches its apogee when the bending force is
perpendicular to the velocity.
This is slowest point in the orbit
One second later the bending forces is accelerating the object
back toward the earth, bending force increases (as object gets
closer)
Example
if gravity all of a sudden disappears, what would happen to object:
Answer: It would fly away from the earth on a linear trajectory.
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Lorenz point
L1 is the Lorenz 1: point between earth and moon such that
gravity cancels out (much closer to the moon than the earth)
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Lorenz point
L1 is the Lorenz 1: point between earth and moon such that
gravity cancels out (much closer to the moon than the earth)
Common error: astronauts in orbit are weightless. Not true.
If a person stood atop a mountain that was 300 km tall
(orbital altitude) they would still weigh about 91.2 percent of
what they would on earth.
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Lorenz point
L1 is the Lorenz 1: point between earth and moon such that
gravity cancels out (much closer to the moon than the earth)
Common error: astronauts in orbit are weightless. Not true.
If a person stood atop a mountain that was 300 km tall
(orbital altitude) they would still weigh about 91.2 percent of
what they would on earth.
Famous zero gravity effect in LEO is not technically correct;
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Lorenz point
L1 is the Lorenz 1: point between earth and moon such that
gravity cancels out (much closer to the moon than the earth)
Common error: astronauts in orbit are weightless. Not true.
If a person stood atop a mountain that was 300 km tall
(orbital altitude) they would still weigh about 91.2 percent of
what they would on earth.
Famous zero gravity effect in LEO is not technically correct;
Due tocontinuous free fall around the earth
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Lorenz point
L1 is the Lorenz 1: point between earth and moon such that
gravity cancels out (much closer to the moon than the earth)
Common error: astronauts in orbit are weightless. Not true.
If a person stood atop a mountain that was 300 km tall
(orbital altitude) they would still weigh about 91.2 percent of
what they would on earth.
Famous zero gravity effect in LEO is not technically correct;
Due tocontinuous free fall around the earth
Entry interface: 121.92 km (400,000 ft) arbitrary
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Lorenz point
L1 is the Lorenz 1: point between earth and moon such that
gravity cancels out (much closer to the moon than the earth)
Common error: astronauts in orbit are weightless. Not true.
If a person stood atop a mountain that was 300 km tall
(orbital altitude) they would still weigh about 91.2 percent of
what they would on earth.
Famous zero gravity effect in LEO is not technically correct;
Due tocontinuous free fall around the earth
Entry interface: 121.92 km (400,000 ft) arbitrary
Example
How is the entry interface defined?
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Lorenz point
L1 is the Lorenz 1: point between earth and moon such that
gravity cancels out (much closer to the moon than the earth)
Common error: astronauts in orbit are weightless. Not true.
If a person stood atop a mountain that was 300 km tall
(orbital altitude) they would still weigh about 91.2 percent of
what they would on earth.
Famous zero gravity effect in LEO is not technically correct;
Due tocontinuous free fall around the earth
Entry interface: 121.92 km (400,000 ft) arbitrary
Example
How is the entry interface defined?
Answer: Pressure drops to zero.
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Cost of fuel to get into orbit
Newtons Third Law: for every action, there is a reaction
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Cost of fuel to get into orbit
Newtons Third Law: for every action, there is a reaction
Contrary to popular belief, rockets do not get into space by
pushing against outside air.
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Cost of fuel to get into orbit
Newtons Third Law: for every action, there is a reaction
Contrary to popular belief, rockets do not get into space by
pushing against outside air.
How rockets worked not clear even as late as last century
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Cost of fuel to get into orbit
Newtons Third Law: for every action, there is a reaction
Contrary to popular belief, rockets do not get into space by
pushing against outside air.
How rockets worked not clear even as late as last century
Engineers discussed why fire boats in East River could not
remain still when pumping water
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Cost of fuel to get into orbit
Newtons Third Law: for every action, there is a reaction
Contrary to popular belief, rockets do not get into space by
pushing against outside air.
How rockets worked not clear even as late as last century
Engineers discussed why fire boats in East River could not
remain still when pumping water
Outside air hinders forward motion. This is why cheaper for
rocket to take off vertically
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Cost of fuel to get into orbit
Newtons Third Law: for every action, there is a reaction
Contrary to popular belief, rockets do not get into space by
pushing against outside air.
How rockets worked not clear even as late as last century
Engineers discussed why fire boats in East River could not
remain still when pumping water
Outside air hinders forward motion. This is why cheaper for
rocket to take off vertically
Example
Why are rockets are more efficient in space than in atmosphere
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Cost of fuel to get into orbit
Newtons Third Law: for every action, there is a reaction
Contrary to popular belief, rockets do not get into space by
pushing against outside air.
How rockets worked not clear even as late as last century
Engineers discussed why fire boats in East River could not
remain still when pumping water
Outside air hinders forward motion. This is why cheaper for
rocket to take off vertically
Example
Why are rockets are more efficient in space than in atmosphere
Answer: Think of nozzle. Now put a plate over the nozzle
and rocket stops.
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Cost of fuel to get into orbit
Newtons Third Law: for every action, there is a reaction
Contrary to popular belief, rockets do not get into space by
pushing against outside air.
How rockets worked not clear even as late as last century
Engineers discussed why fire boats in East River could not
remain still when pumping water
Outside air hinders forward motion. This is why cheaper for
rocket to take off vertically
Example
Why are rockets are more efficient in space than in atmosphere
Answer: Think of nozzle. Now put a plate over the nozzle
and rocket stops.
Air acts like partial plate.
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Orbits
Practically speaking an orbit must be outside a planet’s
atmosphere.
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Orbits
Practically speaking an orbit must be outside a planet’s
atmosphere.
Any satellite that reaches orbital velocity within Earth’s
atmosphere will be melted by the heat created as it collides
with air molecules.
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Orbits
Practically speaking an orbit must be outside a planet’s
atmosphere.
Any satellite that reaches orbital velocity within Earth’s
atmosphere will be melted by the heat created as it collides
with air molecules.
Must be above entry interface to avoid aerocapture
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Orbits
Practically speaking an orbit must be outside a planet’s
atmosphere.
Any satellite that reaches orbital velocity within Earth’s
atmosphere will be melted by the heat created as it collides
with air molecules.
Must be above entry interface to avoid aerocapture
Entry interface is 400,000 ft (pressure drops to zero) arbitrary
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Orbital velocities decrease with altitude
Orbital velocities
Sea level
200k
1000k
10,000k
384,400k (Moon)
Ortt cloud
Earth
7,905 m/s
7,784 m/s
7,350 m/s
4,933 m/s
1010 m/s∗
200 m/s
Lunar
1,680 m/s
1,591 m/s
1,338 m/s
Note: 4 times speed of a passenger jet
Source: Sellers et al. Understanding Space
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Orbits
Orbits are either circular or elliptical
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Orbits
Orbits are either circular or elliptical
An ellipse is defined by two foci and the sum of the distances
between the foci and the perimeter is always a constant
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Orbits
Orbits are either circular or elliptical
An ellipse is defined by two foci and the sum of the distances
between the foci and the perimeter is always a constant
The flatness of the ellipse is determined by the eccentricity.
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Orbits
Orbits are either circular or elliptical
An ellipse is defined by two foci and the sum of the distances
between the foci and the perimeter is always a constant
The flatness of the ellipse is determined by the eccentricity.
The greater the eccentricity the more elongated is the ellipse.
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Orbits
Orbits are either circular or elliptical
An ellipse is defined by two foci and the sum of the distances
between the foci and the perimeter is always a constant
The flatness of the ellipse is determined by the eccentricity.
The greater the eccentricity the more elongated is the ellipse.
An eccentricity of zero defines a circle
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Orbits
Orbits are either circular or elliptical
An ellipse is defined by two foci and the sum of the distances
between the foci and the perimeter is always a constant
The flatness of the ellipse is determined by the eccentricity.
The greater the eccentricity the more elongated is the ellipse.
An eccentricity of zero defines a circle
An elliptical orbit around the Earth has the center of the
Earth at one focus.
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Orbits
Orbits are either circular or elliptical
An ellipse is defined by two foci and the sum of the distances
between the foci and the perimeter is always a constant
The flatness of the ellipse is determined by the eccentricity.
The greater the eccentricity the more elongated is the ellipse.
An eccentricity of zero defines a circle
An elliptical orbit around the Earth has the center of the
Earth at one focus.
Example
Why are orbits either elliptical or circular.
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Orbits
Orbits are either circular or elliptical
An ellipse is defined by two foci and the sum of the distances
between the foci and the perimeter is always a constant
The flatness of the ellipse is determined by the eccentricity.
The greater the eccentricity the more elongated is the ellipse.
An eccentricity of zero defines a circle
An elliptical orbit around the Earth has the center of the
Earth at one focus.
Example
Why are orbits either elliptical or circular.
Answer: It is a property of the two-body problem
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Ellipse
r1
r2
F2
2c
2a
major axis
The math of an ellipse
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minor axis
2b
C
F1
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Conic sections and eccentricity
Conic sections
Circle
Ellipse
Parabola
Hyperbola
e= 0
0<e<1
e=1
e>1
Source: Sellers et al. Understanding Space
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Orbital geometry
Figure 5-5. Inclination. Inclination, i, deUniversity
of Vermont
scribes the tilt Bill
of Gibson
the orbital
plane
with respect
Thus,
plane wi
orbit's o
The fo
used to d
f . Before
its piece
except it
Earth. So
along th
Now
node" (o
normally
Can Humans Fly in Space
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were ellipses but couldn't say why. We've just shown why:
oving in a gravitational field must follow one of the conic
the case of planets or spacecraft in orbit, this path is an ellipse
-hich is just a special case of an ellipse).
we know orbits must follow conic section paths, we can look
_'s to describe the size and shape of an orbit.
-ts
ve're mainly interested in spacecraft orbits, which we know
let's look closer at elliptical geometry. Using Figure
as a
s define some important geometrical parameters for an ellipse.
-'0
R = spacecraft's position vector, measured from
v
Earth's center
V
=
spacecraft's velocity vector
F and F' = primary and vacant foci of the ellipse
Rp
=
radius of perigee (closest approach)
Ra = radius of apogee (farthest approach)
1 - - - - - - - - 2c
Ra
2a - - - - - - - - - - 1
2a
=
major axis
2b
=
minor axis
2c
=
distance between the foci
a
=
semimajor axis
b
=
semiminor axis
v
=
true anomaly
4>
= flight-path angle
Geometry of an Elliptical Orbit. With these parameters, we completely define the size and shape of the orbit.
the radius from the focus of the ellipse (in this case, Earth's
er) to the spacecraft
d F' are the primary (occupied) and vacant (unoccupied) foci.
's center is at the occupied focus.
. the radius of periapsis (radius of the closest approach of the
cecraft to the occupied focus); it's called the radius of perigee when
orbit is around Earth
is the radius ofapoapsis (radius of the farthest approach of the
cecraft to the occupied focus); it's called the radius of apogee when
orbit is around Earth
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Classical Orbital Elements
a
e
i
Ω
ω
t
ν
Semi major axis
measures the orbit’s size
Eccentricity
describes the orbit’s shape
Inclination
measures orbital tilt
Right ascension of ascending node
Swivel angle of equatorial crossing
Argument of perigee
closest approach point
time
time to perigee
True anomaly
Angle from perigee to spacecraft’s position
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Classic Orbital Elements (COEs). Here we show four of the six COEs. We use
visualize an orbit and locate a spacecraft in it. The other two COEs, semimajor
eccentricity, e, specify the siZe and shape of an orbit.
Summary of Classic Orbital Elements.
Name
Description
Range of Values
Undefined
Semimajor axis
Size
Depends on the
conic section
Never
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Orbits
Note that period or altitude at perigee is not considered part
of the classical set.
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Orbits
Note that period or altitude at perigee is not considered part
of the classical set.
Inclination-orbits cannot be centered over either the northern
or southern hemisphere
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Orbits
Note that period or altitude at perigee is not considered part
of the classical set.
Inclination-orbits cannot be centered over either the northern
or southern hemisphere
They must either be aligned with the equator or inclined
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Orbits
Note that period or altitude at perigee is not considered part
of the classical set.
Inclination-orbits cannot be centered over either the northern
or southern hemisphere
They must either be aligned with the equator or inclined
The lower the orbit the more fuel required for a given
inclination change
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Orbits
Note that period or altitude at perigee is not considered part
of the classical set.
Inclination-orbits cannot be centered over either the northern
or southern hemisphere
They must either be aligned with the equator or inclined
The lower the orbit the more fuel required for a given
inclination change
Practically a satellite can only change a few degrees in LEO
Bill Gibson
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Orbits
Note that period or altitude at perigee is not considered part
of the classical set.
Inclination-orbits cannot be centered over either the northern
or southern hemisphere
They must either be aligned with the equator or inclined
The lower the orbit the more fuel required for a given
inclination change
Practically a satellite can only change a few degrees in LEO
Can change 30 degrees in geosynchronous orbit
Bill Gibson
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Orbits
Note that period or altitude at perigee is not considered part
of the classical set.
Inclination-orbits cannot be centered over either the northern
or southern hemisphere
They must either be aligned with the equator or inclined
The lower the orbit the more fuel required for a given
inclination change
Practically a satellite can only change a few degrees in LEO
Can change 30 degrees in geosynchronous orbit
Example
If the inclination is zero, the projection path makes over the Earth
is a straight line. What if the inclination is not zero?
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Orbits
Note that period or altitude at perigee is not considered part
of the classical set.
Inclination-orbits cannot be centered over either the northern
or southern hemisphere
They must either be aligned with the equator or inclined
The lower the orbit the more fuel required for a given
inclination change
Practically a satellite can only change a few degrees in LEO
Can change 30 degrees in geosynchronous orbit
Example
If the inclination is zero, the projection path makes over the Earth
is a straight line. What if the inclination is not zero?
Answer: The path is a sin wave
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Projection path
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Why does this happen?
In the figure below, the diagonal line represents the inclined
orbit viewed from space
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Why does this happen?
In the figure below, the diagonal line represents the inclined
orbit viewed from space
Break up time so that the vehicle moves along the orbital
path as shown by the arrow in the top lefthand figure.
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Why does this happen?
In the figure below, the diagonal line represents the inclined
orbit viewed from space
Break up time so that the vehicle moves along the orbital
path as shown by the arrow in the top lefthand figure.
If the Earth didn’t rotate a sequence of arrows would follow
the diagonal line.
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Why does this happen?
In the figure below, the diagonal line represents the inclined
orbit viewed from space
Break up time so that the vehicle moves along the orbital
path as shown by the arrow in the top lefthand figure.
If the Earth didn’t rotate a sequence of arrows would follow
the diagonal line.
But the Earth rotates counterclockwise from above.
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Why does this happen?
In the figure below, the diagonal line represents the inclined
orbit viewed from space
Break up time so that the vehicle moves along the orbital
path as shown by the arrow in the top lefthand figure.
If the Earth didn’t rotate a sequence of arrows would follow
the diagonal line.
But the Earth rotates counterclockwise from above.
Hence the first arrow shifts to the right with the rotation.
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Why does this happen?
In the figure below, the diagonal line represents the inclined
orbit viewed from space
Break up time so that the vehicle moves along the orbital
path as shown by the arrow in the top lefthand figure.
If the Earth didn’t rotate a sequence of arrows would follow
the diagonal line.
But the Earth rotates counterclockwise from above.
Hence the first arrow shifts to the right with the rotation.
The next arrow is on the trajectory again.
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Why does this happen?
In the figure below, the diagonal line represents the inclined
orbit viewed from space
Break up time so that the vehicle moves along the orbital
path as shown by the arrow in the top lefthand figure.
If the Earth didn’t rotate a sequence of arrows would follow
the diagonal line.
But the Earth rotates counterclockwise from above.
Hence the first arrow shifts to the right with the rotation.
The next arrow is on the trajectory again.
Repeat these steps to draw a sinusoidal line as shown in the
lower left.
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Other orbits
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Geostationary orbits
Geostationary orbits must have zero inclination otherwise they
are call geosynchronous
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Geostationary orbits
Geostationary orbits must have zero inclination otherwise they
are call geosynchronous
They are good for communications satellites because they
require no relays.
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Geostationary orbits
Geostationary orbits must have zero inclination otherwise they
are call geosynchronous
They are good for communications satellites because they
require no relays.
But they are expensive to put in place and keep there.
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Geostationary orbits
Geostationary orbits must have zero inclination otherwise they
are call geosynchronous
They are good for communications satellites because they
require no relays.
But they are expensive to put in place and keep there.
They are so far away, there can be signal delay
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Geostationary orbits
Geostationary orbits must have zero inclination otherwise they
are call geosynchronous
They are good for communications satellites because they
require no relays.
But they are expensive to put in place and keep there.
They are so far away, there can be signal delay
Trace out a figure 8 path on the surface of the Earth as shown
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Geostationary orbits
Geostationary orbits must have zero inclination otherwise they
are call geosynchronous
They are good for communications satellites because they
require no relays.
But they are expensive to put in place and keep there.
They are so far away, there can be signal delay
Trace out a figure 8 path on the surface of the Earth as shown
Example
Which orbit is best for satellite imaging?
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Geostationary orbits
Geostationary orbits must have zero inclination otherwise they
are call geosynchronous
They are good for communications satellites because they
require no relays.
But they are expensive to put in place and keep there.
They are so far away, there can be signal delay
Trace out a figure 8 path on the surface of the Earth as shown
Example
Which orbit is best for satellite imaging?
Answer: Non geosynchronous, but polar
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Polar path
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Johannes Kepler (1610)
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First Law
Planets move in elliptic orbits with the sun at one focus
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First Law
Planets move in elliptic orbits with the sun at one focus
Formation of the sun created too much energy for circular
orbits.
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First Law
Planets move in elliptic orbits with the sun at one focus
Formation of the sun created too much energy for circular
orbits.
All the planets are in the same plane about the sun.
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First Law
Planets move in elliptic orbits with the sun at one focus
Formation of the sun created too much energy for circular
orbits.
All the planets are in the same plane about the sun.
Earth’s orbit has an eccentricity of 0.017 (almost circular)
but Pluto has an eccentricity of 0.248
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First Law
Planets move in elliptic orbits with the sun at one focus
Formation of the sun created too much energy for circular
orbits.
All the planets are in the same plane about the sun.
Earth’s orbit has an eccentricity of 0.017 (almost circular)
but Pluto has an eccentricity of 0.248
The nomenclature for orbits depends on what body the
satellite is
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First Law
Planets move in elliptic orbits with the sun at one focus
Formation of the sun created too much energy for circular
orbits.
All the planets are in the same plane about the sun.
Earth’s orbit has an eccentricity of 0.017 (almost circular)
but Pluto has an eccentricity of 0.248
The nomenclature for orbits depends on what body the
satellite is
Example
Does a perigee exist for the Moon?
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First Law
Planets move in elliptic orbits with the sun at one focus
Formation of the sun created too much energy for circular
orbits.
All the planets are in the same plane about the sun.
Earth’s orbit has an eccentricity of 0.017 (almost circular)
but Pluto has an eccentricity of 0.248
The nomenclature for orbits depends on what body the
satellite is
Example
Does a perigee exist for the Moon?
Answer: No!
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Here are the terms...
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Earth
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Kepler’s 2nd Law
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Perigee velocities
Satellites go faster at perigee
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Perigee velocities
Satellites go faster at perigee
Speed at perigee and apogee
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Kepler’s 3rd Law
Period of orbit depends on altitude
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Kepler’s Laws determine orbits
First: Planets move in elliptic orbits with the sun at one focus
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Kepler’s Laws determine orbits
First: Planets move in elliptic orbits with the sun at one focus
Second: Equal areas in equal times
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Kepler’s Laws determine orbits
First: Planets move in elliptic orbits with the sun at one focus
Second: Equal areas in equal times
Third: Period of orbit depends on altitude.
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Kepler’s Laws determine orbits
First: Planets move in elliptic orbits with the sun at one focus
Second: Equal areas in equal times
Third: Period of orbit depends on altitude.
Example
What are the implications of these three laws:
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Kepler’s Laws determine orbits
First: Planets move in elliptic orbits with the sun at one focus
Second: Equal areas in equal times
Third: Period of orbit depends on altitude.
Example
What are the implications of these three laws:
Answer: 1. No other satellite motion possible
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Kepler’s Laws determine orbits
First: Planets move in elliptic orbits with the sun at one focus
Second: Equal areas in equal times
Third: Period of orbit depends on altitude.
Example
What are the implications of these three laws:
Answer: 1. No other satellite motion possible
2. Speed increases dramatically at perigee
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Kepler’s Laws determine orbits
First: Planets move in elliptic orbits with the sun at one focus
Second: Equal areas in equal times
Third: Period of orbit depends on altitude.
Example
What are the implications of these three laws:
Answer: 1. No other satellite motion possible
2. Speed increases dramatically at perigee
3. Period is proportional to mean distance from primary focus
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Orbital maneuvering
Thrust corresponds to ∆v
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Orbital maneuvering
Thrust corresponds to ∆v
Higher orbits require more energy to get there
Bill Gibson
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Orbital maneuvering
Thrust corresponds to ∆v
Higher orbits require more energy to get there
Changing inclination is difficult; changing altitude is much
easier and requires a low ∆v .
Bill Gibson
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Orbital maneuvering
Thrust corresponds to ∆v
Higher orbits require more energy to get there
Changing inclination is difficult; changing altitude is much
easier and requires a low ∆v .
Where thrust ∆v is added makes a big difference.
Bill Gibson
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Can Humans Fly in Space
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Orbital maneuvering
Thrust corresponds to ∆v
Higher orbits require more energy to get there
Changing inclination is difficult; changing altitude is much
easier and requires a low ∆v .
Where thrust ∆v is added makes a big difference.
If the initial orbit is circular then adding thrust will increase
the eccentricity
Bill Gibson
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Can Humans Fly in Space
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Orbital maneuvering
Thrust corresponds to ∆v
Higher orbits require more energy to get there
Changing inclination is difficult; changing altitude is much
easier and requires a low ∆v .
Where thrust ∆v is added makes a big difference.
If the initial orbit is circular then adding thrust will increase
the eccentricity
Example
What is a ∆v budget?
Bill Gibson
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Orbital maneuvering
Thrust corresponds to ∆v
Higher orbits require more energy to get there
Changing inclination is difficult; changing altitude is much
easier and requires a low ∆v .
Where thrust ∆v is added makes a big difference.
If the initial orbit is circular then adding thrust will increase
the eccentricity
Example
What is a ∆v budget?
Answer: Check out next slide...
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∆v
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∆v budget
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What does ∆v do?
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Make orbit more elliptical
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Circularize orbit
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Hohmann transfer-low to higher orbit
Adding thrust a perigee will increase the eccentricity
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Hohmann transfer-low to higher orbit
Adding thrust a perigee will increase the eccentricity
Adding thrust at the apogee will decrease the eccentricity
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Hohmann transfer-low to higher orbit
Adding thrust a perigee will increase the eccentricity
Adding thrust at the apogee will decrease the eccentricity
Hohmann transfer combines both for most efficient orbital
transfer
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Hohmann transfer-low to higher orbit
Adding thrust a perigee will increase the eccentricity
Adding thrust at the apogee will decrease the eccentricity
Hohmann transfer combines both for most efficient orbital
transfer
Start with a burn at perigee-elongate the orbit
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Hohmann transfer-low to higher orbit
Adding thrust a perigee will increase the eccentricity
Adding thrust at the apogee will decrease the eccentricity
Hohmann transfer combines both for most efficient orbital
transfer
Start with a burn at perigee-elongate the orbit
Give apogee kick to re-circularize the orbit at new altitude
Bill Gibson
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Can Humans Fly in Space
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Hohmann transfer-low to higher orbit
Adding thrust a perigee will increase the eccentricity
Adding thrust at the apogee will decrease the eccentricity
Hohmann transfer combines both for most efficient orbital
transfer
Start with a burn at perigee-elongate the orbit
Give apogee kick to re-circularize the orbit at new altitude
Both ∆v burns in same direction
Bill Gibson
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Can Humans Fly in Space
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Hohmann transfer-low to higher orbit
Adding thrust a perigee will increase the eccentricity
Adding thrust at the apogee will decrease the eccentricity
Hohmann transfer combines both for most efficient orbital
transfer
Start with a burn at perigee-elongate the orbit
Give apogee kick to re-circularize the orbit at new altitude
Both ∆v burns in same direction
Example
What happens to the period?
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Hohmann transfer-low to higher orbit
Adding thrust a perigee will increase the eccentricity
Adding thrust at the apogee will decrease the eccentricity
Hohmann transfer combines both for most efficient orbital
transfer
Start with a burn at perigee-elongate the orbit
Give apogee kick to re-circularize the orbit at new altitude
Both ∆v burns in same direction
Example
What happens to the period?
Answer: It increases by Kepler’s third law!
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Examples
Example
What happens to vehicle velocity in first burn?
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Examples
Example
What happens to vehicle velocity in first burn?
Answer: It decreases in transfer orbit
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Examples
Example
What happens to vehicle velocity in first burn?
Answer: It decreases in transfer orbit
Example
What happens to vehicle velocity in second burn?
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Examples
Example
What happens to vehicle velocity in first burn?
Answer: It decreases in transfer orbit
Example
What happens to vehicle velocity in second burn?
Answer: It increases
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Examples
Example
What happens to vehicle velocity in first burn?
Answer: It decreases in transfer orbit
Example
What happens to vehicle velocity in second burn?
Answer: It increases
Example
How would you get from a higher to lower orbit?
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Examples
Example
What happens to vehicle velocity in first burn?
Answer: It decreases in transfer orbit
Example
What happens to vehicle velocity in second burn?
Answer: It increases
Example
How would you get from a higher to lower orbit?
Answer: Start with a retro burn at apogee...increase
eccentricity gives a lower altitude at perigee. Retro burn at
perigee to recircularize.
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Hohmann transfer
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Fast transfer
Most efficient from point of view of fuel burn
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Fast transfer
Most efficient from point of view of fuel burn
Takes a long time to execute
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Fast transfer
Most efficient from point of view of fuel burn
Takes a long time to execute
Another option?
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Rendezvous uses Hohmann transfer orbit
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Lunar Trajectory
Also combinations of Hohmann transfer trajectories.
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Lunar Trajectory
Also combinations of Hohmann transfer trajectories.
Spacecraft must first reach escape velocity to leave Earth’s
gravity.
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Lunar Trajectory
Also combinations of Hohmann transfer trajectories.
Spacecraft must first reach escape velocity to leave Earth’s
gravity.
Must have enough energy to break out of the orbit
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Lunar Trajectory
Also combinations of Hohmann transfer trajectories.
Spacecraft must first reach escape velocity to leave Earth’s
gravity.
Must have enough energy to break out of the orbit
As spacecraft reaches Lagrange point, L1, does a retroburn to
allow it to be captured by the gravitational field of the moon.
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Lunar Trajectory
Also combinations of Hohmann transfer trajectories.
Spacecraft must first reach escape velocity to leave Earth’s
gravity.
Must have enough energy to break out of the orbit
As spacecraft reaches Lagrange point, L1, does a retroburn to
allow it to be captured by the gravitational field of the moon.
Without this retrofire, the spacecraft will pass the moon or
crash into it (or orbit the sun)
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Lunar Trajectory
Also combinations of Hohmann transfer trajectories.
Spacecraft must first reach escape velocity to leave Earth’s
gravity.
Must have enough energy to break out of the orbit
As spacecraft reaches Lagrange point, L1, does a retroburn to
allow it to be captured by the gravitational field of the moon.
Without this retrofire, the spacecraft will pass the moon or
crash into it (or orbit the sun)
Example
Is this what was used in Apollo?
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Lunar Trajectory
Also combinations of Hohmann transfer trajectories.
Spacecraft must first reach escape velocity to leave Earth’s
gravity.
Must have enough energy to break out of the orbit
As spacecraft reaches Lagrange point, L1, does a retroburn to
allow it to be captured by the gravitational field of the moon.
Without this retrofire, the spacecraft will pass the moon or
crash into it (or orbit the sun)
Example
Is this what was used in Apollo?
Answer: Yes! There is really no other way.
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Leave Earth
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Lunar
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Other Missions
The same idea would be used to go to Mars or other planets
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Other Missions
The same idea would be used to go to Mars or other planets
Leave Earth’s gravitational field and begin to orbit the Sun
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Other Missions
The same idea would be used to go to Mars or other planets
Leave Earth’s gravitational field and begin to orbit the Sun
Perform a Hohmann transfer
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Other Missions
The same idea would be used to go to Mars or other planets
Leave Earth’s gravitational field and begin to orbit the Sun
Perform a Hohmann transfer
Rendezvous with Mars
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Other Missions
The same idea would be used to go to Mars or other planets
Leave Earth’s gravitational field and begin to orbit the Sun
Perform a Hohmann transfer
Rendezvous with Mars
When the spacecraft approaches Mars (or any other planet) it
must retrofire to become captured in the Martian orbit
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Other Missions
The same idea would be used to go to Mars or other planets
Leave Earth’s gravitational field and begin to orbit the Sun
Perform a Hohmann transfer
Rendezvous with Mars
When the spacecraft approaches Mars (or any other planet) it
must retrofire to become captured in the Martian orbit
Example
Which way would you want to approach Mars?
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Other Missions
The same idea would be used to go to Mars or other planets
Leave Earth’s gravitational field and begin to orbit the Sun
Perform a Hohmann transfer
Rendezvous with Mars
When the spacecraft approaches Mars (or any other planet) it
must retrofire to become captured in the Martian orbit
Example
Which way would you want to approach Mars?
Answer: It is cheaper to perform this maneuver by
approaching Mars from the same direction rather than from
the opposite direction, since that latter would require a larger
retro burn.
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Orbit Sun
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Mars
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Motion of Satellites
To find the path of a Satellites set the gravitational force equal
to the centripetal force of the orbiting body, vehicle or planet
GMe m
mv 2
=
r2
r
where G = universal gravitational constant
G = 6.67259 × 10−11 m3 kg −1 s −2
and Me is the mass of the earth
Me = 5.983 × 1024 kg
Bill Gibson
University of Vermont
(1)
Can Humans Fly in Space
Costs
Orbital velocity
Now can calculate velocity by noting that the angle θ
subtends an arc of ds = rd θ
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Orbital velocity
Now can calculate velocity by noting that the angle θ
subtends an arc of ds = rd θ
Here θ is measured in radians.
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Orbital velocity
Now can calculate velocity by noting that the angle θ
subtends an arc of ds = rd θ
Here θ is measured in radians.
Radians is a measure of an angle and is related to the
circumference of a circle 2πr .
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Orbital velocity
Now can calculate velocity by noting that the angle θ
subtends an arc of ds = rd θ
Here θ is measured in radians.
Radians is a measure of an angle and is related to the
circumference of a circle 2πr .
Note that if the circle were of radius 1, the total arc length
would be 2π
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Orbital velocity
Now can calculate velocity by noting that the angle θ
subtends an arc of ds = rd θ
Here θ is measured in radians.
Radians is a measure of an angle and is related to the
circumference of a circle 2πr .
Note that if the circle were of radius 1, the total arc length
would be 2π
Corresponds in a unit circle to an arc length of 2π
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Orbital velocity
Now can calculate velocity by noting that the angle θ
subtends an arc of ds = rd θ
Here θ is measured in radians.
Radians is a measure of an angle and is related to the
circumference of a circle 2πr .
Note that if the circle were of radius 1, the total arc length
would be 2π
Corresponds in a unit circle to an arc length of 2π
When angles are measured in radians, the relationship is
one-to-one
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Orbital velocity
Now can calculate velocity by noting that the angle θ
subtends an arc of ds = rd θ
Here θ is measured in radians.
Radians is a measure of an angle and is related to the
circumference of a circle 2πr .
Note that if the circle were of radius 1, the total arc length
would be 2π
Corresponds in a unit circle to an arc length of 2π
When angles are measured in radians, the relationship is
one-to-one
An angle of x rad corresponds to exactly an arc length of x
when the radius is 1 (rx when the radius is r )
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
ds
dt
= r ddtθ
ds = rd!"
d!"
M
r
Velocity of the spacecraft (as measured by instruments on
board) will be equal to the rate of change of the angle
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Orbital velocity
Take the angular velocity equation
v =r
Bill Gibson
dθ
dt
University of Vermont
Can Humans Fly in Space
Costs
Orbital velocity
Take the angular velocity equation
v =r
dθ
dt
Square it
v 2 = r 2(
Bill Gibson
dθ 2
)
dt
University of Vermont
Can Humans Fly in Space
Costs
Orbital velocity
Take the angular velocity equation
v =r
dθ
dt
Square it
v 2 = r 2(
Plug into equation
GMe m
r2
=
mv 2
r
dθ 2
)
dt
above, we can write:
GMm
mv 2
dθ
=
= mr ( )2
2
r
r
dt
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Orbital velocity
Cancel the mass, m, of the spacecraft
GM = r 3 (
Bill Gibson
dθ 2
)
dt
University of Vermont
Can Humans Fly in Space
Costs
Orbital velocity
Cancel the mass, m, of the spacecraft
GM = r 3 (
dθ 2
)
dt
The orbital time, T , is then given by the period of revolution
of the spacecraft.
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Orbital velocity
Cancel the mass, m, of the spacecraft
GM = r 3 (
dθ 2
)
dt
The orbital time, T , is then given by the period of revolution
of the spacecraft.
T is the length of time it takes for one orbit (about 90
minutes in LEO)
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Orbital velocity
Cancel the mass, m, of the spacecraft
GM = r 3 (
dθ 2
)
dt
The orbital time, T , is then given by the period of revolution
of the spacecraft.
T is the length of time it takes for one orbit (about 90
minutes in LEO)
For one orbit
dθ
dθ
= 2π/T and ( )2 = 4π 2 /T 2
dt
dt
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Orbital velocity
Substituting
GM = r 3
Bill Gibson
4π 2
T2
University of Vermont
Can Humans Fly in Space
Costs
Orbital velocity
Substituting
4π 2
T2
The orbital time, T , is then given by the period of revolution
of the spacecraft orbit (about 90 minutes in LEO)
r
r
2
4π
r
T = r3
= 2πr
GM
GM
GM = r 3
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Orbital velocity
Substituting
4π 2
T2
The orbital time, T , is then given by the period of revolution
of the spacecraft orbit (about 90 minutes in LEO)
r
r
2
4π
r
T = r3
= 2πr
GM
GM
GM = r 3
This is basic equation of satellite motion and holds for
elliptical orbits when r is the semi-major axis
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Orbital velocity summary
The greater the distance, the slower the orbital speed
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Orbital velocity summary
The greater the distance, the slower the orbital speed
Orbital speed of a satellite depends on velocity of the satellite
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Orbital velocity summary
The greater the distance, the slower the orbital speed
Orbital speed of a satellite depends on velocity of the satellite
And mass of the body the satellite orbits
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Orbital velocity summary
The greater the distance, the slower the orbital speed
Orbital speed of a satellite depends on velocity of the satellite
And mass of the body the satellite orbits
Not the satellite mass, density, size or other factors
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Orbital velocity summary
The greater the distance, the slower the orbital speed
Orbital speed of a satellite depends on velocity of the satellite
And mass of the body the satellite orbits
Not the satellite mass, density, size or other factors
Note that G,M and π are all given numbers
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Orbital velocity summary
The greater the distance, the slower the orbital speed
Orbital speed of a satellite depends on velocity of the satellite
And mass of the body the satellite orbits
Not the satellite mass, density, size or other factors
Note that G,M and π are all given numbers
Will be given on tests
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Orbital velocity summary
The greater the distance, the slower the orbital speed
Orbital speed of a satellite depends on velocity of the satellite
And mass of the body the satellite orbits
Not the satellite mass, density, size or other factors
Note that G,M and π are all given numbers
Will be given on tests
Keep in mind that the r is measured from the center of the
massive body
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Orbital Energy
New way to look at orbits
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Orbital Energy
New way to look at orbits
Different from gravitational attraction = centripetal
acceleration
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Orbital Energy
New way to look at orbits
Different from gravitational attraction = centripetal
acceleration
Defined as the sum of kinetic and potential
E = Ek + Ep
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Orbital Energy
New way to look at orbits
Different from gravitational attraction = centripetal
acceleration
Defined as the sum of kinetic and potential
E = Ek + Ep
Kinetic =
mv 2
2
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Orbital Energy
New way to look at orbits
Different from gravitational attraction = centripetal
acceleration
Defined as the sum of kinetic and potential
E = Ek + Ep
Kinetic =
mv 2
2
Potential force × distance,r , gives = − GMm
r
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Orbital Energy
New way to look at orbits
Different from gravitational attraction = centripetal
acceleration
Defined as the sum of kinetic and potential
E = Ek + Ep
Kinetic =
mv 2
2
Potential force × distance,r , gives = − GMm
r
Potential energy enters with negative sign since it opposes
kinetic energy...also weaker the higher you go.
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
What is this telling us?
Think of system of Earth/spacecraft as one unit
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
What is this telling us?
Think of system of Earth/spacecraft as one unit
Unit can be hot or cold
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
What is this telling us?
Think of system of Earth/spacecraft as one unit
Unit can be hot or cold
Faster the spacecraft the hotter
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
What is this telling us?
Think of system of Earth/spacecraft as one unit
Unit can be hot or cold
Faster the spacecraft the hotter
Higher the spacecraft the hotter
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
What is this telling us?
Think of system of Earth/spacecraft as one unit
Unit can be hot or cold
Faster the spacecraft the hotter
Higher the spacecraft the hotter
Gravity acts to cool the system...slow it down
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
What is this telling us?
Think of system of Earth/spacecraft as one unit
Unit can be hot or cold
Faster the spacecraft the hotter
Higher the spacecraft the hotter
Gravity acts to cool the system...slow it down
But gravity is weaker as spacecraft gets higher
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
What is this telling us?
Think of system of Earth/spacecraft as one unit
Unit can be hot or cold
Faster the spacecraft the hotter
Higher the spacecraft the hotter
Gravity acts to cool the system...slow it down
But gravity is weaker as spacecraft gets higher
Example
What is escape velocity?
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
What is this telling us?
Think of system of Earth/spacecraft as one unit
Unit can be hot or cold
Faster the spacecraft the hotter
Higher the spacecraft the hotter
Gravity acts to cool the system...slow it down
But gravity is weaker as spacecraft gets higher
Example
What is escape velocity?
Answer: The energy level (heat) that will allow spacecraft to
escape gravitational cooling
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Orbits from the perspective of energy
If E < 0 vehicle possesses insufficient kinetic energy to allow
it to escape the planet.
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Orbits from the perspective of energy
If E < 0 vehicle possesses insufficient kinetic energy to allow
it to escape the planet.
Then have familiar elliptical, circular or degraded orbit
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Orbits from the perspective of energy
If E < 0 vehicle possesses insufficient kinetic energy to allow
it to escape the planet.
Then have familiar elliptical, circular or degraded orbit
This is just as discussed from the perspective of balance of
gravitational force and centripetal acceleration
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Orbits from the perspective of energy
If E < 0 vehicle possesses insufficient kinetic energy to allow
it to escape the planet.
Then have familiar elliptical, circular or degraded orbit
This is just as discussed from the perspective of balance of
gravitational force and centripetal acceleration
When E = 0 we have just enough energy to move escape the
massive body
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Orbits from the perspective of energy
If E < 0 vehicle possesses insufficient kinetic energy to allow
it to escape the planet.
Then have familiar elliptical, circular or degraded orbit
This is just as discussed from the perspective of balance of
gravitational force and centripetal acceleration
When E = 0 we have just enough energy to move escape the
massive body
E =
mv 2
2
−
GMm
r
=0
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Calculating escape velocity
E =
mv 2
2
−
GMm
r
=0
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Calculating escape velocity
mv 2
2
− GMm
=0
r
First note that m cancels
E =
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Calculating escape velocity
mv 2
2
− GMm
=0
r
First note that
q m cancels
Gives v = 2GM
re
E =
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Calculating escape velocity
mv 2
2
− GMm
=0
r
First note that
q m cancels
Gives v = 2GM
re
E =
Example
Calculate escape velocity for Earth noting that GM = 4 × 1014 and
the radius of Earth in meters as re = 6.378 × 106
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Calculating escape velocity
mv 2
2
− GMm
=0
r
First note that
q m cancels
Gives v = 2GM
re
E =
Example
Calculate escape velocity for Earth noting that GM = 4 × 1014 and
the radius of Earth in meters as re = 6.378 × 106
Answer: v = 11, 189m/s. This is the ∆v in m/s to escape
Earth’s gravity well–huge number!
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Calculating escape velocity
Note that this is at the surface of the Earth on the equator
and neglects air resistance
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Calculating escape velocity
Note that this is at the surface of the Earth on the equator
and neglects air resistance
So only an a lower bound
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Calculating escape velocity
Note that this is at the surface of the Earth on the equator
and neglects air resistance
So only an a lower bound
In reality it will take a higher ∆v than this!
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Calculating escape velocity
Note that this is at the surface of the Earth on the equator
and neglects air resistance
So only an a lower bound
In reality it will take a higher ∆v than this!
Now want to start a business launching customers to escape
velocity?
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Calculating escape velocity
Note that this is at the surface of the Earth on the equator
and neglects air resistance
So only an a lower bound
In reality it will take a higher ∆v than this!
Now want to start a business launching customers to escape
velocity?
How much would we have to charge them?
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Calculating escape velocity
Note that this is at the surface of the Earth on the equator
and neglects air resistance
So only an a lower bound
In reality it will take a higher ∆v than this!
Now want to start a business launching customers to escape
velocity?
How much would we have to charge them?
Example
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Calculating escape velocity
Note that this is at the surface of the Earth on the equator
and neglects air resistance
So only an a lower bound
In reality it will take a higher ∆v than this!
Now want to start a business launching customers to escape
velocity?
How much would we have to charge them?
Example
Answer: Calculate energy required to for this ∆v
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Launch costs
Energy 12 mv 2
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Launch costs
Energy 12 mv 2
So for one kg we have v 2 /2 = (11, 189)2 m2 /s 2
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Launch costs
Energy 12 mv 2
So for one kg we have v 2 /2 = (11, 189)2 m2 /s 2
= 6.2 × 107 = 62mn joules
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Launch costs
Energy 12 mv 2
So for one kg we have v 2 /2 = (11, 189)2 m2 /s 2
= 6.2 × 107 = 62mn joules
Joule is a measure of energy = work of one Newton
accelerates through one meter distance in the direction of the
force.
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Launch costs
Energy 12 mv 2
So for one kg we have v 2 /2 = (11, 189)2 m2 /s 2
= 6.2 × 107 = 62mn joules
Joule is a measure of energy = work of one Newton
accelerates through one meter distance in the direction of the
force.
Example
How much does a joule of energy cost?
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Launch costs
Energy 12 mv 2
So for one kg we have v 2 /2 = (11, 189)2 m2 /s 2
= 6.2 × 107 = 62mn joules
Joule is a measure of energy = work of one Newton
accelerates through one meter distance in the direction of the
force.
Example
How much does a joule of energy cost?
Answer: 1 joule = 1 watt-second. So multiply this by 1000
to kilowatts and by 3600 to get kWh. In VT we pay about 10
cents per kWh. So we can buy a million 3.6 mn joules for a
dime...36 mn for a dollar!
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Launch costs
Doesn’t look so bad...
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Launch costs
Doesn’t look so bad...
We need a 62.6 mn joules and we can get 36 mn for a dollar
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Launch costs
Doesn’t look so bad...
We need a 62.6 mn joules and we can get 36 mn for a dollar
So 62.6/36 = $1.74...not bad!
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Launch costs
Doesn’t look so bad...
We need a 62.6 mn joules and we can get 36 mn for a dollar
So 62.6/36 = $1.74...not bad!
Example
What is wrong with this calculation?
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Launch costs
Doesn’t look so bad...
We need a 62.6 mn joules and we can get 36 mn for a dollar
So 62.6/36 = $1.74...not bad!
Example
What is wrong with this calculation?
Answer: Several things:
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Launch costs
Doesn’t look so bad...
We need a 62.6 mn joules and we can get 36 mn for a dollar
So 62.6/36 = $1.74...not bad!
Example
What is wrong with this calculation?
Answer: Several things:
Our tourist must weigh only one kilogram.
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Launch costs
Doesn’t look so bad...
We need a 62.6 mn joules and we can get 36 mn for a dollar
So 62.6/36 = $1.74...not bad!
Example
What is wrong with this calculation?
Answer: Several things:
Our tourist must weigh only one kilogram.
Must not object to going into space without a vehicle or
clothes for that matter–just blast him off.
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Launch costs
Doesn’t look so bad...
We need a 62.6 mn joules and we can get 36 mn for a dollar
So 62.6/36 = $1.74...not bad!
Example
What is wrong with this calculation?
Answer: Several things:
Our tourist must weigh only one kilogram.
Must not object to going into space without a vehicle or
clothes for that matter–just blast him off.
Can take no fuel along.
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Launch costs
Doesn’t look so bad...
We need a 62.6 mn joules and we can get 36 mn for a dollar
So 62.6/36 = $1.74...not bad!
Example
What is wrong with this calculation?
Answer: Several things:
Our tourist must weigh only one kilogram.
Must not object to going into space without a vehicle or
clothes for that matter–just blast him off.
Can take no fuel along.
Also remember this is a lower bound for ∆v
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
SS mission
SS orbiter, external tank and SRBs weigh about 200 tons
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
SS mission
SS orbiter, external tank and SRBs weigh about 200 tons
At $1,738 per ton, still a deal: $348k
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
SS mission
SS orbiter, external tank and SRBs weigh about 200 tons
At $1,738 per ton, still a deal: $348k
Example
Why does it cost NASA 600mn to launch the shuttle? Vastly
overpriced?
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
SS mission
SS orbiter, external tank and SRBs weigh about 200 tons
At $1,738 per ton, still a deal: $348k
Example
Why does it cost NASA 600mn to launch the shuttle? Vastly
overpriced?
Answer: Does not count fixed costs only variable costs. Also
liquid hydrogen and oxygen fuel costs much more than
electricity!
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
More realistic mission
Tourist would never come back
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
More realistic mission
Tourist would never come back
Would never survive
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
More realistic mission
Tourist would never come back
Would never survive
Could take any fuel to save himself
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
More realistic mission
Tourist would never come back
Would never survive
Could take any fuel to save himself
Business would face catastrophic liability lawsuits
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
More realistic mission
Tourist would never come back
Would never survive
Could take any fuel to save himself
Business would face catastrophic liability lawsuits
Not a valid business plan!
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
More realistic mission
Tourist would never come back
Would never survive
Could take any fuel to save himself
Business would face catastrophic liability lawsuits
Not a valid business plan!
Example
What would be better?
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
More realistic mission
Tourist would never come back
Would never survive
Could take any fuel to save himself
Business would face catastrophic liability lawsuits
Not a valid business plan!
Example
What would be better?
Answer: Richard Branson’s idea of using a futuristic
spaceplane to take tourists to the edge of space, about 100
km or so (62 miles).
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
More realistic mission
Tourist would never come back
Would never survive
Could take any fuel to save himself
Business would face catastrophic liability lawsuits
Not a valid business plan!
Example
What would be better?
Answer: Richard Branson’s idea of using a futuristic
spaceplane to take tourists to the edge of space, about 100
km or so (62 miles).
This is a sub-orbital flight.
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
∆v to altitude
Energy required to lift a vehicle from re to re + h where h is
height above surface is
GM
∆v 2
GM
−
=
re
re + h
2
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
∆v to altitude
Energy required to lift a vehicle from re to re + h where h is
height above surface is
GM
∆v 2
GM
−
=
re
re + h
2
This is the same as
2GM
h
= ∆v 2
(re + h)re
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
∆v to altitude
Energy required to lift a vehicle from re to re + h where h is
height above surface is
GM
∆v 2
GM
−
=
re
re + h
2
This is the same as
2GM
h
= ∆v 2
(re + h)re
Example
What is the ∆v for 100 km?
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
∆v to altitude
Energy required to lift a vehicle from re to re + h where h is
height above surface is
GM
∆v 2
GM
−
=
re
re + h
2
This is the same as
2GM
h
= ∆v 2
(re + h)re
Example
What is the ∆v for 100 km?
Answer: 1, 390m/s
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
∆v and altitude
Highest Jet
X-Prize
Weather balloon
LEO
GSO
Chandra
L5
Altitude h
km
h
miles
∆v
m/sec
16
100
161
300
40,000
133,000
385,630
10
62
100
186
24,855
82,645
239,626
561
1,390
1,755
2,371
10,391
10,930
11,097
Computational notes: ∆v =
q
= 6.67 × 10−11
m3 kg −2 s −1 , M = 5.98 × 1024 kg , re = 6.378 × 106 m
Bill Gibson
2GM
,G
re2 /h+re
University of Vermont
Can Humans Fly in Space
Costs
Escape velocity
Once the rocket has reached the desired orbital distance, must
impart a horizontal velocity.
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Escape velocity
Once the rocket has reached the desired orbital distance, must
impart a horizontal velocity.
Must have sufficient fuel for tangential thrust.
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Escape velocity
Once the rocket has reached the desired orbital distance, must
impart a horizontal velocity.
Must have sufficient fuel for tangential thrust.
If tangential velocity is exact, the orbit will be circular, if not
then elliptical
12000
385,630
133,000
40,000
10000
8000
6000
4000
2000
300
161
100
16
0
0.E+00
5.E+04
1.E+05
2.E+05
2.E+05
Bill Gibson
3.E+05
3.E+05
4.E+05
4.E+05
University of Vermont
Can Humans Fly in Space
Costs
LEO
To get to LEO (300 km) need to go
mv 2 /(re + 300, 000) = 7732m/s
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
LEO
To get to LEO (300 km) need to go
mv 2 /(re + 300, 000) = 7732m/s
7732(m/s )2
2
= 29.892 mn joules
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
LEO
To get to LEO (300 km) need to go
mv 2 /(re + 300, 000) = 7732m/s
7732(m/s )2
2
= 29.892 mn joules
Since we can buy 36 million joules for a dollar
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
LEO
To get to LEO (300 km) need to go
mv 2 /(re + 300, 000) = 7732m/s
7732(m/s )2
2
= 29.892 mn joules
Since we can buy 36 million joules for a dollar
The physical cost of launch is less that a dollar a kg!
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
LEO
To get to LEO (300 km) need to go
mv 2 /(re + 300, 000) = 7732m/s
7732(m/s )2
2
= 29.892 mn joules
Since we can buy 36 million joules for a dollar
The physical cost of launch is less that a dollar a kg!
This is what keeps the space community going!
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
LEO
To get to LEO (300 km) need to go
mv 2 /(re + 300, 000) = 7732m/s
7732(m/s )2
2
= 29.892 mn joules
Since we can buy 36 million joules for a dollar
The physical cost of launch is less that a dollar a kg!
This is what keeps the space community going!
Example
Why can’t this physical limit be approximated?
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
LEO
To get to LEO (300 km) need to go
mv 2 /(re + 300, 000) = 7732m/s
7732(m/s )2
2
= 29.892 mn joules
Since we can buy 36 million joules for a dollar
The physical cost of launch is less that a dollar a kg!
This is what keeps the space community going!
Example
Why can’t this physical limit be approximated?
Answer: For many reasons...
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Rocket equation
at time 0
velocity = v
mass = m
at time t
velocity v + delta v
delta m
mass = m - delta m
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Rocket equation
Start with mass moving at velocity v
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Rocket equation
Start with mass moving at velocity v
Ejects a mass ∆M during a time interval ∆t
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Rocket equation
Start with mass moving at velocity v
Ejects a mass ∆M during a time interval ∆t
This is what makes the rocket move
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Rocket equation
Start with mass moving at velocity v
Ejects a mass ∆M during a time interval ∆t
This is what makes the rocket move
Use Newton’s second law noting that mass is not constant
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Rocket equation
Start with mass moving at velocity v
Ejects a mass ∆M during a time interval ∆t
This is what makes the rocket move
Use Newton’s second law noting that mass is not constant
Mass includes the mass of the vehicle and mass of the
propellant
∆(Mv )
F =
∆t
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Rocket equation
At a moment in time, we have the change in momentum as
the sum of the rocket plus the exhaust:
(M − ∆M )(v + ∆v ) + c∆M
− Mv = 0
{z
} | {z } |{z}
|
rocket
Bill Gibson
exhaust
initial
University of Vermont
Can Humans Fly in Space
Costs
Rocket equation
At a moment in time, we have the change in momentum as
the sum of the rocket plus the exhaust:
(M − ∆M )(v + ∆v ) + c∆M
− Mv = 0
{z
} | {z } |{z}
|
rocket
exhaust
initial
Multiply out
Mv − v ∆M + ∆vM + ∆v ∆M + c∆M − Mv = 0
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Rocket equation
At a moment in time, we have the change in momentum as
the sum of the rocket plus the exhaust:
(M − ∆M )(v + ∆v ) + c∆M
− Mv = 0
{z
} | {z } |{z}
|
rocket
exhaust
initial
Multiply out
Mv − v ∆M + ∆vM + ∆v ∆M + c∆M − Mv = 0
Example
Canceling terms and noting that ∆v ∆M is approximately zero,
what do we have?
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Rocket equation
At a moment in time, we have the change in momentum as
the sum of the rocket plus the exhaust:
(M − ∆M )(v + ∆v ) + c∆M
− Mv = 0
{z
} | {z } |{z}
|
rocket
exhaust
initial
Multiply out
Mv − v ∆M + ∆vM + ∆v ∆M + c∆M − Mv = 0
Example
Canceling terms and noting that ∆v ∆M is approximately zero,
what do we have?
∆M
Answer: M ∆v
∆t = (c − v ) ∆t
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Thrust equation
M ∆v
∆t = T , thrust
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Thrust equation
M ∆v
∆t = T , thrust
c − v = C relative exhaust velocity
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Thrust equation
M ∆v
∆t = T , thrust
c − v = C relative exhaust velocity
Thrust increases with mass or propellant ejected
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Thrust equation
M ∆v
∆t = T , thrust
c − v = C relative exhaust velocity
Thrust increases with mass or propellant ejected
And speed of propellant ejected
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Thrust equation
M ∆v
∆t = T , thrust
c − v = C relative exhaust velocity
Thrust increases with mass or propellant ejected
And speed of propellant ejected
Example
Consider a rocket that weighs 30,000 kg when fueled up on the
launching pad. At burnout weighs 10,000 kg after 30 seconds.
Gases are exhausted at a relative velocity of 1524 m/sec. What is
the average thrust, T at take-off?
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Thrust equation
M ∆v
∆t = T , thrust
c − v = C relative exhaust velocity
Thrust increases with mass or propellant ejected
And speed of propellant ejected
Example
Consider a rocket that weighs 30,000 kg when fueled up on the
launching pad. At burnout weighs 10,000 kg after 30 seconds.
Gases are exhausted at a relative velocity of 1524 m/sec. What is
the average thrust, T at take-off?
Answer:
T = 1524
m 20000 kg
m
(
)
= 1, 016, 000 2 kg = 1 × 106 N
s
30
s
s
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Specific impulse
Rocket engines rated according to specific impulse (measured
in seconds)
C = gIsp
where g is acceleration of gravity
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Specific impulse
Rocket engines rated according to specific impulse (measured
in seconds)
C = gIsp
where g is acceleration of gravity
Example
What are typical Isp ?
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Specific impulse
Rocket engines rated according to specific impulse (measured
in seconds)
C = gIsp
where g is acceleration of gravity
Example
What are typical Isp ?
Answer: Chemical rockets have 200s for kerosene and O2 to
450s for H2 and O2 For orbit-to-orbit transfer, have much
higher Isp available: 900s for nuclear and 2, 000s to 20, 000s
for electrical rockets
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Rocket equation
From
M∆v
∆t
= C ∆M
∆t
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Rocket equation
From
M∆v
∆t
Can write
= C ∆M
∆t
∆v
C
=z =
∆M
M
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Rocket equation
From
M∆v
∆t
Can write
= C ∆M
∆t
∆v
C
∆M
M
ez =
=z =
Now use hat rules!
ejected propellant.
M̂ =
Bill Gibson
M +P
M
where P is mass of
University of Vermont
Can Humans Fly in Space
Costs
Rocket equation
From
M∆v
∆t
Can write
= C ∆M
∆t
∆v
C
∆M
M
ez =
=z =
Now use hat rules!
ejected propellant.
M̂ =
M +P
M
where P is mass of
Example
M + P=100 ton rocket all fueled up. Now the mission requires a
Deltav = 9500m/s. Specific impulse is 450s. How much of the
rocket must be lost in propellant ejected?
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Rocket equation
From
M∆v
∆t
Can write
= C ∆M
∆t
∆v
C
∆M
M
ez =
=z =
Now use hat rules!
ejected propellant.
M̂ =
M +P
M
where P is mass of
Example
M + P=100 ton rocket all fueled up. Now the mission requires a
Deltav = 9500m/s. Specific impulse is 450s. How much of the
rocket must be lost in propellant ejected?
Answer: e 9500/[(9.8)450] = 8.62 =
88.41 tons is ejected propellant!
Bill Gibson
100
M
gives M = 11.59 So
University of Vermont
Can Humans Fly in Space
Costs
Mission impossible
What factors could make the mission become completely
infeasible?
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Mission impossible
What factors could make the mission become completely
infeasible?
A higher ∆v ?
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Mission impossible
What factors could make the mission become completely
infeasible?
A higher ∆v ?
A lower specific impulse?
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Mission impossible
What factors could make the mission become completely
infeasible?
A higher ∆v ?
A lower specific impulse?
Could a rocket fly with only one percent of its weight as
structure mass?
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Mission impossible
What factors could make the mission become completely
infeasible?
A higher ∆v ?
A lower specific impulse?
Could a rocket fly with only one percent of its weight as
structure mass?
The payload is counted in the structural mass...usually 10
percent. Here it is only 1.16 tons
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Mission impossible
What factors could make the mission become completely
infeasible?
A higher ∆v ?
A lower specific impulse?
Could a rocket fly with only one percent of its weight as
structure mass?
The payload is counted in the structural mass...usually 10
percent. Here it is only 1.16 tons
Example
What is the advantage of staging?
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Mission impossible
What factors could make the mission become completely
infeasible?
A higher ∆v ?
A lower specific impulse?
Could a rocket fly with only one percent of its weight as
structure mass?
The payload is counted in the structural mass...usually 10
percent. Here it is only 1.16 tons
Example
What is the advantage of staging?
Answer: Reduces the structural mass
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Hohmann transfer
Restrict information to orbits in a plane (coplanar)
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Hohmann transfer
Restrict information to orbits in a plane (coplanar)
Velocity changes are are tangent to the initial and final orbits
(changes velocity but not magnitude).
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Hohmann transfer
Restrict information to orbits in a plane (coplanar)
Velocity changes are are tangent to the initial and final orbits
(changes velocity but not magnitude).
Burns are short (2-5 min) compared with length of orbit.
Treat them as if they were instantaneous.
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Hohmann transfer
Restrict information to orbits in a plane (coplanar)
Velocity changes are are tangent to the initial and final orbits
(changes velocity but not magnitude).
Burns are short (2-5 min) compared with length of orbit.
Treat them as if they were instantaneous.
Coapsidal orbits: two elliptical orbits have their major axes
aligned.
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Hohmann transfer
Restrict information to orbits in a plane (coplanar)
Velocity changes are are tangent to the initial and final orbits
(changes velocity but not magnitude).
Burns are short (2-5 min) compared with length of orbit.
Treat them as if they were instantaneous.
Coapsidal orbits: two elliptical orbits have their major axes
aligned.
Whenever we add or subtract ∆V , we change the orbit’s
specific mechanical energy and hence its size or or semi-major
axis.
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Hohmann transfer
Restrict information to orbits in a plane (coplanar)
Velocity changes are are tangent to the initial and final orbits
(changes velocity but not magnitude).
Burns are short (2-5 min) compared with length of orbit.
Treat them as if they were instantaneous.
Coapsidal orbits: two elliptical orbits have their major axes
aligned.
Whenever we add or subtract ∆V , we change the orbit’s
specific mechanical energy and hence its size or or semi-major
axis.
Example
How do we calculate the burns?
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Hohmann transfer
Restrict information to orbits in a plane (coplanar)
Velocity changes are are tangent to the initial and final orbits
(changes velocity but not magnitude).
Burns are short (2-5 min) compared with length of orbit.
Treat them as if they were instantaneous.
Coapsidal orbits: two elliptical orbits have their major axes
aligned.
Whenever we add or subtract ∆V , we change the orbit’s
specific mechanical energy and hence its size or or semi-major
axis.
Example
How do we calculate the burns?
Answer: It is a two-step procedure!
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Hohmann transfer
Start with Earth orbit and set the energy equal to zero
0 =
Bill Gibson
V 2 GMsun
−
2
r
University of Vermont
Can Humans Fly in Space
Costs
Hohmann transfer
Start with Earth orbit and set the energy equal to zero
0 =
V 2 GMsun
−
2
r
Conservation of energy states that the sum of the kinetic
energy and the potential energy of a particle remains constant
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Hohmann transfer
Start with Earth orbit and set the energy equal to zero
0 =
V 2 GMsun
−
2
r
Conservation of energy states that the sum of the kinetic
energy and the potential energy of a particle remains constant
GMs un = 1.327 × 1011 km3 /s 2
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Hohmann transfer
Start with Earth orbit and set the energy equal to zero
0 =
V 2 GMsun
−
2
r
Conservation of energy states that the sum of the kinetic
energy and the potential energy of a particle remains constant
GMs un = 1.327 × 1011 km3 /s 2
From Kepler’s Law (this conservation of angular momentum).
rp vp = ra va
where ra = radius at aphelion and rp at perihelion
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Hohmann transfer
The kinetic energy
KE =
Bill Gibson
mv 2
2
University of Vermont
Can Humans Fly in Space
Costs
Hohmann transfer
The kinetic energy
KE =
mv 2
2
potential energy of gravity
PE = −
Bill Gibson
GMm
r
University of Vermont
Can Humans Fly in Space
Costs
Hohmann transfer
The kinetic energy
KE =
mv 2
2
potential energy of gravity
PE = −
GMm
r
At the two points, apogee and perigee of the orbit, we have
conservation of energy
PE1 + KE1 = PE2 + KE2
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Hohmann transfer
Substituting
mvp2 GMm
mva2 GMm
−
=
−
2
rp
2
ra
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Hohmann transfer
Substituting
mvp2 GMm
mva2 GMm
−
=
−
2
rp
2
ra
Next cancel out the mass and rearrange the terms:
vp2 − va2 = 2GM (
Bill Gibson
1
1
− )
rp
ra
University of Vermont
Can Humans Fly in Space
Costs
Hohmann transfer
Substituting
mvp2 GMm
mva2 GMm
−
=
−
2
rp
2
ra
Next cancel out the mass and rearrange the terms:
vp2 − va2 = 2GM (
1
1
− )
rp
ra
Substitute Kepler’s second law: vp =
ra va
rp
ra2 va2
1
1
− va2 = 2GM ( − )
rp2
rp
ra
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Hohmann transfer
Simplify to find
va2
=
=
2GM ( r1p −
2
( rra2
p
− 1)
2GMrp (1 −
ra2 (1 −
vp2 =
=
rp
ra )
rp2
)
ra2
2rp GM
(ra + rp ) ra
=
rp
ra )
rp
rp
ra )(1 + ra )
2GMrp (1 −
ra2 (1 −
2GMra
rp (rp + ra )
Va2 = Vp2
Va =
1
ra )
rp2
2rp GM
2GMra
= 2
/
ra
(ra + rp ) ra rp (rp + ra )
rp
Vp
ra
Bill Gibson
University of Vermont
=
2GMrp
ra (ra + rp )
Can Humans Fly in Space
Costs
Hohmann transfer
Simplify to find
va2
=
=
2GM ( r1p −
2
( rra2
p
− 1)
2GMrp (1 −
ra2 (1 −
vp2 =
Done!
=
rp
ra )
rp2
)
ra2
2rp GM
(ra + rp ) ra
=
rp
ra )
rp
rp
ra )(1 + ra )
2GMrp (1 −
ra2 (1 −
2GMra
rp (rp + ra )
Va2 = Vp2
Va =
1
ra )
rp2
2rp GM
2GMra
= 2
/
ra
(ra + rp ) ra rp (rp + ra )
rp
Vp
ra
Bill Gibson
University of Vermont
=
2GMrp
ra (ra + rp )
Can Humans Fly in Space
Costs
Hohmann transfer example
Example: A spacecraft is in a circular earth orbit with an
altitude of 150 miles. Calculate the ∆v 0 s required to change
to a circular orbit with an altitude of 250 miles.
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Hohmann transfer example
Example: A spacecraft is in a circular earth orbit with an
altitude of 150 miles. Calculate the ∆v 0 s required to change
to a circular orbit with an altitude of 250 miles.
Given: r1 = (re + 150)1609.3 = (3960 + 150)1609.3 =
6.6142 × 106 m.
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Hohmann transfer example
Example: A spacecraft is in a circular earth orbit with an
altitude of 150 miles. Calculate the ∆v 0 s required to change
to a circular orbit with an altitude of 250 miles.
Given: r1 = (re + 150)1609.3 = (3960 + 150)1609.3 =
6.6142 × 106 m.
√
Use the basic equation for circular orbits v1 = GM/r .
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Hohmann transfer example
Example: A spacecraft is in a circular earth orbit with an
altitude of 150 miles. Calculate the ∆v 0 s required to change
to a circular orbit with an altitude of 250 miles.
Given: r1 = (re + 150)1609.3 = (3960 + 150)1609.3 =
6.6142 × 106 m.
√
Use the basic equation for circular orbits v1 = GM/r .
For the lower orbit the speed is:
= [(4 × 1014 )
m3
m
/(6.6142 × 106 m )]0.5 = 7766.0
2
s
s
Bill Gibson
University of Vermont
Can Humans Fly in Space
Costs
Hohmann transfer example
Example: A spacecraft is in a circular earth orbit with an
altitude of 150 miles. Calculate the ∆v 0 s required to change
to a circular orbit with an altitude of 250 miles.
Given: r1 = (re + 150)1609.3 = (3960 + 150)1609.3 =
6.6142 × 106 m.
√
Use the basic equation for circular orbits v1 = GM/r .
For the lower orbit the speed is:
m3
m
/(6.6142 × 106 m )]0.5 = 7766.0
2
s
s
Now do the second higher orbit
= [(4 × 1014 )
= [(4 × 1014
m3
m
/(6.6142 × 106 m )]0.5 = 7673.3
2
s
s
Bill Gibson
University of Vermont